Determine the resulting polynomial:

Question

f, of, x, equals, minus, x, minus, 10
f(x)=−x−10
g, of, x, equals, minus, 10, x, squared, plus, 5, x, minus, 4
g(x)=−10x
2
+5x−4
Find: , f, of, x, dot, g, of, x
Find: f(x)⋅g(x)

Answer 1

To find \( f(x) \cdot g(x) \), we first need to know the explicit forms of the functions \( f(x) \) and \( g(x) \).

Given:

\[ f(x) = -x - 10 \] \[ g(x) = -10x^2 + 5x - 4 \]

Now, we multiply these two polynomials together:

\[ f(x) \cdot g(x) = (-x - 10)(-10x^2 + 5x - 4) \]

We can distribute \( -x - 10 \) to each term in \( g(x) \):

1. Multiply \( -x \) by each term in \( g(x) \): \[ -x \cdot -10x^2 = 10x^3 \] \[ -x \cdot 5x = -5x^2 \] \[ -x \cdot -4 = 4x \]

   So, this yields: \[ 10x^3 - 5x^2 + 4x \]

2. Multiply \( -10 \) by each term in \( g(x) \): \[ -10 \cdot -10x^2 = 100x^2 \] \[ -10 \cdot 5x = -50x \] \[ -10 \cdot -4 = 40 \]

   So, this yields: \[ 100x^2 - 50x + 40 \]

3. Now, combine all these results: \[ f(x) \cdot g(x) = (10x^3 - 5x^2 + 4x) + (100x^2 - 50x + 40) \]

4. Combine like terms: \[ = 10x^3 + (-5x^2 + 100x^2) + (4x - 50x) + 40 \] \[ = 10x^3 + 95x^2 - 46x + 40 \]

Thus, the resulting polynomial \( f(x) \cdot g(x) \) is:

\[ \boxed{10x^3 + 95x^2 - 46x + 40} \]